HW6sol - ORIE 3500/5500 Fall Term 2008 Assignment...

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ORIE 3500/5500 Fall Term 2008 Assignment 6-Solution 1. The joint density of X,Y,Z is f X,Y,Z ( x,y,z ) = f X ( x ) f Y ( y ) f Z ( z ) = ± 1 if 0 x,y,z 1 0 otherwise . (a) We have P ( X > Y + Z ) = Z [ x>y + z ] f X,Y,Z ( x,y,z ) dxdydz = Z 1 0 Z 1 - z 0 Z 1 y + z 1 dxdydz = Z 1 0 Z 1 - z 0 (1 - y - z ) dydz = Z 1 0 (1 - z - (1 - z ) 2 / 2 - z (1 - z )) dz = Z 1 0 (1 - z ) 2 / 2 dz = (1 / 2) Z 1 0 w 2 dw = 1 / 6 . (b) We have P ( Y < ZX ) = Z [ y<zx ] f X,Y,Z ( x,y,z ) dydzdx = Z 1 0 Z 1 0 Z zx 0 1 dydzdx = Z 1 0 Z 1 0 zxdzdx = Z 1 0 x (1 / 2) dx = 1 / 4 . 1
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(c) Let W = X + Y . for w [0 , 2], F W ( w ) = P ( W w ) = Z min( w, 1) 0 Z min( w - x, 1) 0 1 .dydx = ± R w 0 R ( w - x ) 0 dx if w < 1 R w - 1 0 1 dx + R 1 w - 1 ( w - x ) dx if w 1 = ± w 2 / 2 if w < 1 2 w - 1 - w 2 / 2 if w 1 and hence f W ( w ) = F 0 W ( w ) = w if 0 w 1 2 - w if 1 w 2 0 otherwise . 2. We know f Y ( y ) = ± 1 if 0 y 1 0 otherwise ,f X | Y ( x | y ) = ± 1 2(1 - y ) if - (1 - y ) x (1 - y ) 0 otherwise . (a) We have f X ( x ) = Z f Y ( y ) f X | Y ( x | y ) dy = ± R 1 - x 0 1 2(1 - y ) dy if x > 0 R 1+ x 0 1 2(1 - y ) dy if x 0 = ± - (1 / 2) log | x | , if - 1 x 1 0 otherwise .
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This note was uploaded on 12/04/2008 for the course STSCI 6000 taught by Professor Turnbull during the Fall '08 term at Cornell University (Engineering School).

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HW6sol - ORIE 3500/5500 Fall Term 2008 Assignment...

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